If, |x+5|<1/2, then show that|x^2-25|<21/4 Could someone help me to solve the problem?

1 Answer
Mar 6, 2018

Kindly refer to the Explanation.

Explanation:

"Prerequisites : "(1) : |a+b| le |a|+|b|, (2) : |ab|=|a||b|.

"By (2), "|x^2-25|=|(x+5)(x-5)|=|x+5||x-5|.

"Now, "|x+5| lt 1/2............[Given].

Multiplying by |x-5| gt 0, we get,

|x+5||x-5| lt 1/2|x-5|............(ast1).

But, |x-5|=|(x+5)+(-10)|,

le |x+5|+|(-10)|...[because, (1)],

=|x+5|+10 lt 1/2+10........[because," Given]".

rArr |x-5| le 21/2............(ast2).

(ast1) and (ast2) rArr |x+5||x-5| lt 1/2*21/2, or,

|x^2-25| lt 21/4, as desired!